Where does God tell us that the “rest (of the Bible) is subservient to the Gospels?”
By His need to send His Son and not communicate the Word and the New Law through surrogates and prophets.
And he said to them, "Go into all the world and preach the gospel to the whole creation." - Mark 16:15
And he said to them, "O foolish men, and slow of heart to believe all that the prophets have spoken! Was it not necessary that the Christ should suffer these things and enter into his glory?" And beginning with Moses and all the prophets, he interpreted to them in all the scriptures the things concerning himself. - Luke 24:25-27
In the first book, O Theophilus, I have dealt with all that Jesus began to do and teach, until the day when he was taken up, after he had given commandments through the Holy Spirit to the apostles whom he had chosen. - Acts 1:1-2
Dei verbum 18:
It is common knowledge that among all the Scriptures, even those of the New Testament, the Gospels have a special preeminence, and rightly so, for they are the principal witness for the life and teaching of the incarnate Word, our savior.
The Church has always and everywhere held and continues to hold that the four Gospels are of apostolic origin. For what the Apostles preached in fulfillment of the commission of Christ, afterwards they themselves and apostolic men, under the inspiration of the divine Spirit, handed on to us in writing: the foundation of faith, namely, the fourfold Gospel, according to Matthew, Mark, Luke and John.